The quoted definition appears to answer your question already.
An 300 ml cup of water is full if and only if it contains exactly 300 ml water. There's no mystery there, if you think of a cup the irony makes no sense because a cup is rigid. The same applies to a tank of petrol, which is normally made from rigid metal.
However, a balloon is flexible and has no one "max volume". You can fill a balloon with fluid or air in the sense that when you hold the balloon, the level of fluid comes up to its brim. It is now "full", and yet you can add additional fluid - it will stretch the balloon, so you will have to apply force to hold it in. But it will still be full, yet with a different volume.
"Full" is not well-defined for flexible containers. It is rather a spectrum. At the end of the spectrum you pump the balloon too much and it tears. If the balloon is actually your stomach, stretching can also be uncomfortable, painful, or lead to vomiting.
The ironic understatement here is exploiting this conceptual difference between full for a rigid object vs. flexible. When you say something is full, people tend to think of a rigid object, because those are simpler and conceptually parsimonious, more common in daily life, and the term "full" is more meaningful for them. But when you use it indirectly for a stomach, it takes some thought to realize that in fact, the container in question is flexible, and it's left unspecified how full it is (ie. whether it's just barely full or very overfull to the point of discomfort). Further thought reveals that had it been just barely full, it would not be worth mention, and had it been overfull, the speaker would have found it inappropriate to say so in polite company. Thus you realize to say "you're full" is actually a complex euphemism for "ugh, I overate, I'm gonna barf".
This is unexpected, and for a certain class of person, a fun discovery. Thus humor is often generated through discovery of a hidden meaning, or surprise. Hence, it is used as ironic understatement - "merely full" as opposed to "stuffed to the point of bursting", and even the possibility of bursting in the first place is understated (as in, it's not stated at all).
Feel free to leave your cheeky comment now with the Mark Twain quote about the frog :)